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 Izv. IMI UdGU, 2020, Volume 55, Pages 33–41 (Mi iimi389)

MATHEMATICS

Asymptotic expansion of a solution of a singularly perturbed optimal control problem with a convex integral quality index, whose terminal part additively depends on slow and fast variables

A. R. Danilina, A. A. Shaburovb

a Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, ul. S. Kovalevskoi, 16, Yekaterinburg, 620108, Russia
b Ural Federal University, ul. Mira, 19, Yekaterinburg, 620002, Russia

Abstract: The paper deals with the problem of optimal control with a Boltz–type quality index over a finite time interval for a linear steady–state control system in the class of piecewise continuous controls with smooth control constraints. In particular, we study the problem of controlling the motion of a system of small mass points under the action of a bounded force. The terminal part of the convex integral quality index additively depends on slow and fast variables, and the integral term is a strictly convex function of control variable. If the system is completely controllable, then the Pontryagin maximum principle is a necessary and sufficient condition for optimality. The main difference between this study and previous works is that the equation contains the zero matrix of fast variables and, thus, the results of A. B. Vasilieva on the asymptotic of the fundamental matrix of a control system are not valid. However, the linear steady–state system satisfies the condition of complete controllability. The article shows that problems of optimal control with a convex integral quality index are more regular than time–optimal problems.

Keywords: optimal control, singularly perturbed problems, asymptotic expansion, small parameter.

DOI: https://doi.org/10.35634/2226-3594-2020-55-03

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Bibliographic databases:

UDC: 517.977
MSC: 49N05, 93C70

Citation: A. R. Danilin, A. A. Shaburov, “Asymptotic expansion of a solution of a singularly perturbed optimal control problem with a convex integral quality index, whose terminal part additively depends on slow and fast variables”, Izv. IMI UdGU, 55 (2020), 33–41

Citation in format AMSBIB
\Bibitem{DanSha20} \by A.~R.~Danilin, A.~A.~Shaburov \paper Asymptotic expansion of a solution of a singularly perturbed optimal control problem with a convex integral quality index, whose terminal part additively depends on slow and fast variables \jour Izv. IMI UdGU \yr 2020 \vol 55 \pages 33--41 \mathnet{http://mi.mathnet.ru/iimi389} \crossref{https://doi.org/10.35634/2226-3594-2020-55-03}